Seminar in Number Theory and Algebraic Geometry

Abstract: Roth's theorem on arithmetic progressions is about how large a subset of {1,2,...,N} can be without containing three numbers in arithmetic progression, i.e. of the form x, x+d, x+2d with d non-zero. The problem was first studied in the 1930s, and has since then shown itself to have lots of interesting connections to other parts of mathematics, such as harmonic analysis and high-dimensional convex geometry. In a 2023 breakthrough, Zander Kelley and Raghu Meka proved much stronger upper bounds for the problem than had previously been known, using a myriad of interesting techniques. In this talk I will give an overview of their argument and some modifications that allowed for some simplifications and generalisations. Based on joint work with Thomas Bloom.